= Conditional cure-weight equations for paired Poisson counts
{title2=$\widehat\theta=n^{-1}\sum_i q_i,\quad\widehat p=\frac{\sum_i y_i}{\sum_i(1-q_i)t_i}$}
Consider an approximate <conditional likelihood> in which a paired total $t_i$ leads to a structural zero with probability $\theta$, and otherwise a <binomial distribution> with success probability $p$. For an observed zero, the posterior cure weight is $q_i=\theta/[\theta+(1-\theta)(1-p)^{t_i}]$; for a positive count it is zero. Differentiating the <log-likelihood> gives the displayed equations at an interior <maximum-likelihood estimator>, with the weights evaluated at that estimator. Recomputing the weights and then updating the parameters is an <expectation-maximization algorithm>. Boundary fits must also be considered. These equations describe a conditional mixture: if cure is instead specified before two <Poisson distributions> are conditioned on their total, its conditional probability generally depends on the total and on the baseline nuisance rate. A zero observed in a short interval is not by itself proof of permanent biological cure.
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